Mathquery

Lioville's Theorem :- Statement :       If a function f(z) is analytic for all limit values of z and is bounded then f(z) is constant . Proof :        Let z₁ , z₂ be any two point of the z-plane the contour C to be a large circle of radius R centred at origin and containing the point z₁ , z₂ . Therefore |r₁|<R and |z₂|<R . Also as f(z) is bounded therefore ∃ a positive M such that f|(z)|≤M for all z.      By Cauchy's Integral formula we have       f(z₁) = 1/2πi ∫ f(z) dz /z-z₁                           c      f(z₂) = 1/2πi ∫ f(z) dz /z-z₂                           c ∴ f(z₁) - f(z₂) = 1/2πi ∫ f(z) dz /z-z₁                                ...